Question:** How many of the 150 smallest positive integers are congruent to 3 (mod 7)?
["Title: How Many of the First 150 Positive Integers Are Congruent to 3 (mod 7)?", "---", "When exploring modular arithmetic in number theory, one common question is: how many numbers in a specific range are congruent to a given integer modulo another number? This article dives into a precise calculation about the integers from 1 to 150 and how many are congruent to 3 modulo 7.", "### What Does “Congruent to 3 mod 7” Mean?", "An integer n is congruent to 3 modulo 7 if it leaves a remainder of 3 when divided by 7. Mathematically, this is written as:\n[\nn \equiv 3 \pmod{7}\n]\nThis means n can be expressed as:\n[\nn = 7k + 3\n]\nfor some integer k.", "### Finding All Such Numbers Among the First 150 Positive Integers", "We want all integers n such that:\n- ( 1 \leq n \leq 150 )\n- ( n = 7k + 3 ) for some non-negative integer k", "Let’s find the smallest and largest k that satisfy this within the range.", "1. Smallest k:\nSet ( k = 0 ):\n[\nn = 7(0) + 3 = 3 \quad \ ext{(within range)}\n]", "2. Largest k such that ( 7k + 3 \leq 150 ):\nSolve:\n[\n7k + 3 \leq 150\n]\n[\n7k \leq 147\n]\n[\nk \leq 21\n]\nSo the largest valid k is 21.", "Thus, k ranges from 0 to 21 inclusive.", "### Counting the Valid Values", "The number of integers k from 0 to 21 is:\n[\n21 - 0 + 1 = 22\n]", "Therefore, there are 22 positive integers between 1 and 150 that are congruent to 3 modulo 7.", "### Examples of These Numbers", "The numbers are:\n3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150", "(Note: 150 = 7×21 + 3, so included.)", "### Summary", "- Total integers from 1 to 150: 150\n- Numbers ≡ 3 mod 7: 22\n- Pattern: All of the form ( 7k + 3 ) with ( k = 0,1,2,\ldots,21 )", "Understanding how many elements satisfy a modular condition helps in number theory, algorithm analysis, and cryptography applications. The first 150 positive integers contain exactly 22 integers congruent to 3 modulo 7.", "---", "Keywords: congruent to 3 mod 7, integers 1 to 150, modular arithmetic, number theory, count multiples, modular residues, mathematical problem solving", "Meta Description:\nDiscover how many of the first 150 positive integers are congruent to 3 modulo 7. Learn the method and see the full list of 22 such numbers.", "---", "Want to master modular counting? Start with small moduli and increase the range—this technique becomes powerful for large datasets!"]
